Lecture 5: 2nd and 3rd Laws of Thermodynamics

2nd law: The entropy of an isolated system never decreases.

Nernst’s Theorem: The entropy of a system at T = 0 is a well-defined constant. For any processes that bring a system at T = 0 from one equilibrium state to

another, S = 0 .

An isolated system, being initially in a non-equilibrium state, will evolve from macropartitions with lower multiplicity (lower probability, lower entropy) to macropartitions with higher multiplicity (higher probability, higher entropy). Once the system reaches the macropartition with the highest multiplicity (highest entropy), it will stay there.

1

,

,...),,(

NVU

NVUST

From S(N,U,V) - to U(N,T,V)

Find (U,V,N,...) – the most challenging step

S (U,V,N,...) = kB ln (U,V,N,...)

Solve for U = f (T,V,N,...)

1

,

,...),,(

NVU

NVUST

Now we can get an (energy) equation of state U = f (T,V,N,...) for any

system for which we have an explicit formula for the multiplicity (entropy)!!

Thus, we’ve bridged the gap between statistical mechanics and

thermodynamics! The recipe:

Measuring Entropy Even if we cannot calculate S, we can still measure it:

For V=const and N=const :T

dTTC

T

Q

T

dUdS V )(

T

V

T

TdTCSTS

0

)()0()(

By heating a cup of water (200g, CV =840 J/K) from 200C to 1000C, we increase its entropy by

J/K 200J/K) 840(373

293

T

TdS

At the same time, the multiplicity of the system is increased by25105.1 e

- in L.6, we’ll see that this equation holds for all reversible (quasi-static) processes (even if V is changed in the process).T

QdS

This is the “thermodynamic” definition of entropy, which Clausius introduced in 1854, long before Boltzmann gave his “statistical” definition S kBln .

An Einstein Solid: from S(N,U) to U(N,T) at high T

NN

N

eU

N

qeUN

),(High temperatures:

(kBT >> , q >>N)

N

ekNUkN

N

eUkNNUS BBB lnlnln),(

U

kN

U

S

TB

1

TkNTNU B),(

- in agreement with the equipartition theorem: the total energy should be ½kBT times the number of degrees of freedom.

To compare with experiment, we can measure the heat capacity:

NVV T

UC

,

dT

PdVdU

dT

QC

the heat capacity at constant volume

BBV kNTkNT

C

- in a nice agreement with experiment

An Einstein Solid: from S(N,U) to U(N,T) at low T

Low temperatures:

(kBT <<, q <<N)Uq

U

eN

q

eNUN

),(

U

eNUkUNS B

ln),(

U

Nkk

U

eNk

U

S

TBBB

lnln1

TkBeNTNU

),,(

- as T 0, the energy goes to zero as expected (Pr. 3.5).

The low-T heat capacity: (more accurate result will be obtained on the basis of the

Debye model of solids)

Tk

BB

B

TkTkV

BBB eTk

NkTk

eNeNT

C

2

2

Nernst’s Theorem (1906)The Third Law of Thermodynamics

T

V

T

TdTCSTS

0

)()0()(

A non-trivial issue – what’s the value of S(0)?

The answer is provided by Q.M. (discreteness of quantum states), it cannot be deduced from the other laws of thermodynamics – thus, the third law:

Nernst’s Theorem: The entropy of a system at T = 0 is a well-defined constant. For any processes that bring a system at T = 0 from one equilibrium state to

another, S = 0 .

This is because a system at T = 0 exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state.

A special case – systems with unique ground state, such as elementary

crystals : S(0) = 0 ( = 1). However, the claim that S(0) is "usually" zero or

negligible is wrong.

In other words, S(T) approaches a finite limit at T = 0 , which does not depend on specifics of processes that brought the system to the T = 0 state.

0 ln 0BS k

Residual Entropy

If S(0) 0, it’s called residual entropy.

1. For compounds, for example, there is frequently a significant amount of entropy that comes from a multiplicity of possible molecular orientations in the crystal (degeneracy of the ground state), even at absolute zero.

Structure of the hexagonal ice: - H atoms, --- - covalent O-H bonds, - - - - weak H bonds. Six H atoms can “tunnel” simultaneously through a potential barrier.

Wang et al, nature, 439, 303 (2006)

Residual Entropy (cont.)

2. Nernst’s Theorem applies to the equilibrium states only!

Glasses aren’t really in equilibrium, the relaxation time – huge. They do not have a well-defined T or CV. Glasses have a particularly large entropy at T = 0.

T

S

liquid

crystal

glass

supercooledliquid

residualentropy

Debenedetti & Stillinger, Nature (2001)

Example (Pr. 3.14, page 97)

For a mole of aluminum, CV = aT + bT3 at T < 50 K (a = 1.35·10-3 J/K2, b = 2.48·10–5 J/K4). The linear term – due to mobile electrons, the cubic term – due to the crystal lattice vibrations. Find S(T) and evaluate the entropy at T = 1K,10 K.

3

0

3

0 3

)()( T

baT

T

TdTbTa

T

TdTCTS

TTV

J/K 1036.1K1J/K 1048.23

1K1J/K 1035.1)1( 334523 KST = 1K

- at low T, nearly all the entropy comes from the mobile electrons

T = 10K J/K 1018.2K10J/K 1048.23

1K10J/K 1035.1)10( 2334523 KS

- most of the entropy comes from lattice vibrations

2123

220

23

3

106.1J/K 1038.1

J/K 1018.2)10( 10

J/K 1038.1

J/K 1035.1)1(

BB k

KS

k

KS

- much less than the # of particles, most degrees of freedom are still frozen out.

C 0 as T 0

T

V

T

TdTCSTS

0

)()0()( 0 as 0)( TTC

For example, let’s fix P :

dTTCQ P )( )(

00

TP

T

T

TdTC

T

Q- finite, CP(0) = 0

Similar, considering V = const - CV(0) = 0 .

Thus, the specific heat must be a function of T. We know that at high T, CV approaches a universal temperature-independent limit that depends on N and # of degrees of freedom. This high - T behavior cannot persist down to T = 0 – quantum effects will come into play.